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6. A variational method for quantitative photoacoustic tomography with piecewise constant coefficients

From the book Variational Methods

  • Elena Beretta , Monika Muszkieta , Wolf Naetar and Otmar Scherzer


We consider the inverse problem of determining spatially heterogeneous absorption and diffusion coefficients μ(x), D(x), from a single measurement of the absorbed energy E(x) = μ(x)u(x), where u satisfies the elliptic partial differential equation

−∇ ⋅ (D(x)∇u(x)) + μ(x)u(x) =0 in Ω ⊂ ℝN .

This problem, which is central in quantitative photoacoustic tomography, is in general ill-posed since it admits an infinite number of solution pairs. Using similar ideas as in [31], we show that when the coefficients μ, D are known to be piecewise constant functions, a unique solution can be obtained. For the numerical determination of μ, D, we suggest a variational method based on an Ambrosio-Tortorelli approximation of a Mumford-Shah-like functional, which we implement numerically and test on simulated two-dimensional data.

© 2016 Walter de Gruyter GmbH, Berlin/Munich/Boston
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